Prove that $1+2^1+2^2+\ldots +2^n=2^{n+1}-1$ using induction

Question

For all integers $n\ge 1$ prove the following statement using mathematical induction.

$$1+2^1+2^2+\ldots +2^n=2^{n+1}-1$$

The first part of the question ask me to prove the base step:

So I set $n=1$ and plugged it in but the answer is not correct (unless I made a silly arithmetic error somewhere).

$2^1=2^{1+1}-1$

$2=2^{2}-1$

$2=4-1$

$2=3$

Am I doing something wrong here?

Answer 1

Your mistake is quite simple: You omitted the initial "$1+\cdots$".

When $n=0$ the identity says $1 = 2 - 1$, i.e. $2^0=2^{0+1}-1$. You can take that to be the base step.

Answer 2

Your LHS should be $1 + \ldots + 2^n = 1 + \ldots + 2^1 = 1 + 2^1 = 3$

Answer 3

This is not an answer for this question but from proof without words.